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You can use this to prove the Cantor-Schröder-Bernstein theorem, which asserts that whenever $A$ injects into $B$ and $B$ injects into $A$, then they are bijective. Namely, suppose that $f:A\to B$ and $g:B\to A$ are both injective functions. If there were a set $X\subset A$ such that $A-X=g[B-f[X]]$, then the function $h=(f\upharpoonright X)\cup (g^{-1}\upharpoonright A-X)$ is a bijection between $A$ and $B$. Such a set $X$ exists by the Knaster-Tarski theorem, since the powerset $P(A)$ is a complete lattice under inclusion and the function $\varphi(X)=A-g[B-f[X]]$ is $\subset$-preserving, since $$X\subset Y\implies f[X]\subset f[Y]\implies B-f[X]\supset B-f[Y]$$ $$\implies A-g[B-f[X]]\subset A-g[B-f[Y]]\implies \varphi(X)\subset\varphi(Y).$$
A fixed point $X=\varphi(x)$ means $A-X=g[B-f[X]]$.

A useful application of Tarski's fixed point theorem is that every supermodular game (mostly games with strategic complementarities) has a smallest and a largest pure strategy Nash equilibrium. For surveys of supermodular games, see here, here, or here. The literature is huge. By a slight modification of the theorem, one can actually show that the set of pure strategy Nash equilibria forms a complete lattice in itself.

In computer science, it is used in the field of denotational semantics and abstract interpretation, where the existence of fixed points can be exploited to guarantee well-defined semantics for a recursive algorithm, see this for an example.